January 8th, 2017, 12:34 PM  #1 
Newbie Joined: Jan 2017 From: los angeles Posts: 2 Thanks: 0  optimization problem
Consider an optimization problem, minimize f(Ax)+x^T x the variable is nvector x. The matrix A has size m by n and rank m. f is not necessarily differentiable or convex. Show that the problem can be formulated as an equivalent problem with m variables, by making change of y = Ax. 
January 26th, 2017, 05:30 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,067 Thanks: 691 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Since $\displaystyle y = Ax, x = A^{1} y$, $\displaystyle A^{1} A = 1$ and $\displaystyle (AB)^T = B^T A^T$ $\displaystyle f(Ax) + x^T x$ $\displaystyle = f(y) + (A^{1} y)^T A^{1} y$ $\displaystyle = f(y) + y^T (A^{1})^T A^{1} y$ This is almost exactly the same as the original equation except that y is the new variable of interest and you have to work out $\displaystyle (A^{1})^T A^{1}$ as the filling of the matrix sandwich Last edited by Benit13; January 26th, 2017 at 05:41 AM. 

Tags 
optimization, problem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Optimization Problem  atari  Calculus  3  March 6th, 2016 07:45 PM 
Optimization Problem .. Please help  fantom2012  Calculus  6  April 25th, 2012 10:41 PM 
need help for optimization problem  latzi  Calculus  1  July 24th, 2009 05:05 PM 
Optimization problem  wonger357  Calculus  1  May 15th, 2009 12:42 PM 
Optimization Problem  a.a  Calculus  0  April 19th, 2008 10:54 PM 